A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Expressing a stochastic integral in terms of an ordinary integral

Let $g_n (t)= \sqrt{2} \cos \big[ (n- \frac{1}{2}) \pi t \big]$ and $h_n (t)= \sqrt{2} \sin \big[ (n- \frac{1}{2}) \pi t \big]$, and let $W$ be a Brownian motion. Then how can we show that ...
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11 views

Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
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1answer
24 views

Convergence of exponential Brownian martingale to zero almost surely

Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian ...
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25 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
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Poisson Processes question

. Let {N(t) : t ≥ 0} be a Poisson process with rate λ > 0. Let Y be a random variable independent of (N(t)), such that Y = 1 with probability 1/2 and Y = −1 with probability 1/2. We define the new ...
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10 views

Poisson Processes - What is the distribution of the number of arrivals $Z$ happening in the random interval of time $[0,T]$?

Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $λ$, and $Z$ represent the number of arrivals in the interval of time $[0,t]$. Let $T$ be a random variable, exponentially distributed with ...
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24 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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26 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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25 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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Point processes that are not Cox?

Can some provide examples of point processes that are not Cox? A Cox process is a doubly stochastic poisson process with random intensity.
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47 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
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27 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
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11 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
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1answer
32 views

Reference request for stochastic process

I studied the book, "Probability with the book, Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal ...
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30 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
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Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
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1answer
33 views

Adaptive gambler's ruin problem

Suppose in the gambler's ruin problem that the probability of winning a bet depends on the gambler's present fortune. Specifically, suppose that $p_{i}$ is the probability that the gambler wins a bet ...
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Men and Women enter a supermarket according to independent poisson process (stochastic process) [on hold]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
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Variance of interarrival time of events [on hold]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
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1answer
55 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
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How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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1answer
31 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
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1answer
51 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
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1answer
35 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
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1answer
31 views

Process adapted to Filtration [on hold]

Here is the definition I have been given : A process $(X_t)$ is adapted to a filtration $(\mathcal F_t)$ if $X_t$ is $F_t$ measurable, for all t > 0 , i.e : $X_t^-1 (\mathcal B)$ belong to ...
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Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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Simple Symmetric Random Walk [on hold]

Use Hint: Show first that for any random variable N with range {0,1,...},
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32 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
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17 views

How to get the PDF for $p(a)$ $p(b)$ by using convolution?

Suppose there are three random variable $a$, $b$, $c$, and the PDF for each are $p(a)\ p(b)$ and $p(c) $ Also, $c$ = $a$ + $b$, $a$ and $b$ are two independent variable. and$$p(c)= \begin{cases} ...
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29 views

Applying the martingale representation theorem

I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. ...
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28 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
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19 views

Finding the expectation and variance of a stochastic process

Let $X_0, \ldots$ be i.i.d. $\mathbb{P}\{X_i = -1\} = \mathbb{P}\{X_i = 1\} = 1 / 2$. Given $a, b \in \mathbb{R}, |b| < 1$, consider the stochastic process $W_k$ defined as $$ W_0 = a X_0\\ W_k = b ...
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Two-state Markov Chains

If I have a two-state Markov chian $V(t)$ with transition probabilities: $P_{00}(t)=(1-\pi) + \pi e^{-\tau t}$ $P_{01}(t)= \pi - \pi e^{-\tau t}$ $P_{10}(t)=(1-\pi) - (1-\pi)e^{-\tau t}$ ...
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33 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...
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Find the expected value ${\mathbb E}(x_k)$

$X_k = X_{k-1} + m_k(x_0=0)$, and pdf(probability density function) of $m_k$ is defined as $$p(m_k)= \begin{cases} m_k+1&-1\le m_k<0\\ -m_k+1&0\le m_k\le1\\ 0&\text{otherwise} ...
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21 views

Finding solution to this stochastic differential equation

Let $W, Z$ be two correlated Brownian motions with $dW\,dZ=\rho\, dt$. We also have the following three processes: \begin{align} dD_t &= rD_t \,dt & & (D_T=1, r>0)\\ dS_t &= rS ...
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Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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Using Central Limit Theorem to show that random walk exits a interval a.s. in finite time.

Let $X_0 = x \in \mathbb{Z}$ and $X_1, X_2, \dots$ are i.i.d. random variables with values in $\{-1,0,1\}$ all with positive probability and $E(X_1) = 0$. Let $\sigma^2 = E(X_1^2)$. Let $S_n = ...
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61 views

Right-continuous supermartingale is almost sure cadlag

Suppose that we have a supermartingale $M$ defined on an underlying filtered probability space $(\Omega,\mathcal{F},P)$. We assume that the index set is defined by $T=\mathbb{R}^+$ and for notation we ...
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30 views

how to determine transient and recurrent state from transition matrix

I wonder how can I determine the transient and recurrent state from transition matrix ? I mean if I have 10 states It would be very hard to draw diagram for them so how to analyse the matrix? For ...
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31 views

Calculating inter-arrival times and arrival times of a Poisson process

For a practice exam in stochastic processes I have to answer the following questions. Let $\{N(t): t\geq 0\}$ be a poisson process with rate $\lambda$. Let $T_n$ denote the n-th inter-arrival time ...
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32 views

Weak convergence of an integral of an exponential of a Wiener process

Suppose $(W_s)_{s \geq 0}$ is a Wiener process. Define $$ V_t := \frac{1}{\sqrt{t}}\int_{0}^{t}\exp(W_s)ds $$ Show that $$ V_t \xrightarrow{t \rightarrow \infty} \sup_{s \in[0, 1]}W_s $$ in ...
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1answer
15 views

Stationary Probabilities: Periodic Case: motivation 2nd attempt.

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? ...
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27 views

Stationary probabilities: periodic case: motivation

For DTMC with $S=\{1,2\}$ and transition probabilities $$P = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$ How do we see that $(P_{00})^{(n)} = 1$ if $n$ is even or $0$ if $n$ is odd ?? ...
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37 views

Lindeberg-Feller CLT follows from Martingale CLT?

I've been studying about central limit theorems, in particular the Lindeberg-Feller CLT and other extensions. In most textbooks and sources online the martingale central limit theorem comes as an ...
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1answer
28 views

Kullback-Leibler divergence when the $Q$ distribution has zero values

For discrete probability distributions $P,Q$, the Kullback-Leibler divergence of $Q$ from $P$ is defined to be $$D_{\mathrm{KL}} ( P \mathop{\|} Q ) = \sum_i P(i) \ln \left( \frac{P(i)}{Q(i)} ...
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1answer
19 views

Period of a Markov Chain: Why is this one aperiodic?

Here is the problem from a stochastic processes book: Consider a Markov Chain on {0,1,2} having transition matrix 0 1 2 0| 0 0 1| 1| 1 0 0| 2|.5 .5 0| ...
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26 views

Optimization of stochastic differential equations

Is there a way to optimize or maximize a set of differential equations. such that each equation is represented by a time series S_((t+1),μ) = μ*(S_(t+1)-S_t) + S_t and μ = 2/(i+1), i=1,...,n. Then I ...
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1answer
34 views

Joint Density Function of uniform and gamma density [closed]

Let $U$ be uniformly distributed over the interval $[0, L]$ where $L$ follows the gamma density $f_L(x) = xe^{-x}$ for $x\ge 0$. What is the joint desnity function of $U$ and $V = L - U$?